585 research outputs found

### On the stochastic Cahn-Hilliard equation with a singular double-well potential

We prove well-posedness and regularity for the stochastic pure Cahn-Hilliard
equation under homogeneous Neumann boundary conditions, with both additive and
multiplicative Wiener noise. In contrast with great part of the literature, the
double-well potential is treated as generally as possible, its convex part
being associated to a multivalued maximal monotone graph everywhere defined on
the real line on which no growth nor smoothness assumptions are assumed. The
regularity result allows to give appropriate sense to the chemical potential
and to write a natural variational formulation of the problem. The proofs are
based on suitable monotonicity and compactness arguments in a generalized
variational framework.Comment: 37 page

### A doubly nonlinear evolution problem related to a model for microwave heating

This paper is concerned with the existence and uniqueness of the solution to
a doubly nonlinear parabolic problem which arises directly from a circuit model
of microwave heating. Beyond the relevance from a physical point of view, the
problem is very interesting also in a mathematical approach: in fact, it
consists of a nonlinear partial differential equation with a further
nonlinearity in the boundary condition. Actually, we are going to prove a
general result: the two nonlinearities are allowed to be maximal monotone
operators and then an existence result will be shown for the resulting problem.Comment: Key words and phrases: nonlinear parabolic equation, nonlinear
boundary condition, existence of solution

### Optimal distributed control of a stochastic Cahn-Hilliard equation

We study an optimal distributed control problem associated to a stochastic
Cahn-Hilliard equation with a classical double-well potential and Wiener
multiplicative noise, where the control is represented by a source-term in the
definition of the chemical potential. By means of probabilistic and analytical
compactness arguments, existence of an optimal control is proved. Then the
linearized system and the corresponding backward adjoint system are analysed
through monotonicity and compactness arguments, and first-order necessary
conditions for optimality are proved.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase
separation, optimal control, linearized state system, adjoint state system,
first-order optimality condition

### Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: a variational approach

We prove existence of invariant measures for the Markovian semigroup
generated by the solution to a parabolic semilinear stochastic PDE whose
nonlinear drift term satisfies only a kind of symmetry condition on its
behavior at infinity, but no restriction on its growth rate is imposed. Thanks
to strong integrability properties of invariant measures $\mu$, solvability of
the associated Kolmogorov equation in $L^1(\mu)$ is then established, and the
infinitesimal generator of the transition semigroup is identified as the
closure of the Kolmogorov operator. A key role is played by a generalized
variational setting.Comment: 32 page

### Fr\'echet differentiability of mild solutions to SPDEs with respect to the initial datum

We establish n-th order Fr\'echet differentiability with respect to the
initial datum of mild solutions to a class of jump-diffusions in Hilbert
spaces. In particular, the coefficients are Lipschitz continuous, but their
derivatives of order higher than one can grow polynomially, and the
(multiplicative) noise sources are a cylindrical Wiener process and a
quasi-left-continuous integer-valued random measure. As preliminary steps, we
prove well-posedness in the mild sense for this class of equations, as well as
first-order G\^ateaux differentiability of their solutions with respect to the
initial datum, extending previous results in several ways. The
differentiability results obtained here are a fundamental step to construct
classical solutions to non-local Kolmogorov equations with sufficiently regular
coefficients by probabilistic means.Comment: 30 pages, no figure

### From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation

A rigorous proof is given for the convergence of the solutions of a viscous
Cahn-Hilliard system to the solution of the regularized version of the
forward-backward parabolic equation, as the coefficient of the diffusive term
goes to 0. Non-homogenous Neumann boundary condition are handled for the
chemical potential and the subdifferential of a possible non-smooth double-well
functional is considered in the equation. An error estimate for the difference
of solutions is also proved in a suitable norm and with a specified rate of
convergence.Comment: Key words and phrases: Cahn-Hilliard system, forward-backward
parabolic equation, viscosity, initial-boundary value problem, asymptotic
analysis, well-posednes

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